The Welsh club were on the verge of dropping out of the Football League altogether back in the 2002/03 season but managed to remain in League Two with a victory over Hull City on the final day of the campaign.
Today at Wembley, Swansea won the Capital One Cup 5-0 over Bradford City and midfielder Britton was sharing his joy as to what the club has achieved in the past 10 years.
"It's been unbelievable the last 10 years. We nearly dropped out of the Football League and now here we are in the Premier League, winning our first major honour and qualifying for Europe," he told Sky Sports 1.
"It was a great day. You've got to give credit to Bradford, they've been brilliant throughout the tournament. Unfortunately there has to be a winner and a loser. Obviously I'm delighted we've won but you've got to feel for the Bradford lads."
Regarding playing in Europe next season, he added: "It'll be great. It will be a great adventure next season for the players and fans alike."